2-1. Dependency Measures
library(reshape2) # melt function
library(ggplot2) # ggplot function
library(pcaPP) # Fast Kendall function
library(energy) # Distance Correlation
library(Hmisc) # Hoeffding's D measure
## Loading required package: lattice
## Loading required package: survival
## Loading required package: Formula
##
## Attaching package: 'Hmisc'
## The following objects are masked from 'package:dplyr':
##
## src, summarize
## The following object is masked from 'package:SeuratObject':
##
## Key
## The following object is masked from 'package:Seurat':
##
## Key
## The following objects are masked from 'package:base':
##
## format.pval, units
library(zebu) # Normalized Mutual Information
# library(minerva) # Maximum Information Coefficient
library(XICOR) # Chatterjee's Coefficient
# library(dHSIC) # Hilbert Schmidt Independence Criterion
library(VineCopula) # Blomqvist's Beta
make_cormat <- function(dat_mat){
matrix_dat <- matrix(nrow = ncol(dat_mat), ncol = ncol(dat_mat))
cor_mat_list <- list()
basic_cor <- c("pearson", "spearman")
# find each of the correlation matrices with Pearson, Spearman, Kendall Correlation Coefficients
for (i in 1:2){
print(i)
cor_mat <- stats::cor(dat_mat, method = basic_cor[i])
cor_mat[upper.tri(cor_mat, diag = T)] <- NA
cor_mat_list[[i]] <- cor_mat
save(cor_mat_list, file = "../temp_cor_data.Rdata")
}
# functions that take matrix or data.frame as input
no_loop_function <- c(pcaPP::cor.fk, Hmisc::hoeffd,
VineCopula::BetaMatrix)
for (i in 3:5){
print(i)
fun <- no_loop_function[[i-2]]
cor_mat <- fun(dat_mat)
if (i == 4){ # Hoeffding's D
cor_mat <- cor_mat$D
}
cor_mat[upper.tri(cor_mat, diag = T)] <- NA
cor_mat_list[[i]] <- cor_mat
save(cor_mat_list, file = "../temp_cor_data.Rdata")
}
# functions that take two variables as input to calculate correlations.
need_loop <- c(zebu::lassie, energy::dcor2d, XICOR::calculateXI)
for (i in 6:8){
print(i)
fun <- need_loop[[i-5]]
cor_mat <- matrix(nrow = ncol(dat_mat),
ncol = ncol(dat_mat))
for (j in 2:ncol(dat_mat)){
for (k in 1:(j-1)){
if (i == 6){
cor_mat[j, k] <- fun(cbind(dat_mat[, j], dat_mat[, k]), continuous=c(1,2), breaks = 6, measure = "npmi")$global
} else {
cor_mat[j, k] <- fun(as.numeric(dat_mat[, j]),
as.numeric(dat_mat[, k]))
}
}
}
cor_mat[upper.tri(cor_mat, diag = T)] <- NA
cor_mat_list[[i]] <- cor_mat
save(cor_mat_list, file = "../temp_cor_data.Rdata")
}
return(cor_mat_list)
}
draw_heatmap <- function(cor_mat){
len <- 5
melted_cormat <- melt(cor_mat)
melted_cormat <- melted_cormat[!is.na(melted_cormat$value),]
break_vec <- round(as.numeric(quantile(melted_cormat$value,
probs = seq(0, 1, length.out = len),
na.rm = T)),
4)
break_vec[1] <- break_vec[1]-1
break_vec[len] <- break_vec[len]+1
melted_cormat$value <- cut(melted_cormat$value, breaks = break_vec)
heatmap_color <- unique(melted_cormat$value)
heatmap <- ggplot(data = melted_cormat, aes(x = Var2, y = Var1, fill = value))+
geom_tile() +
ggplot2::scale_fill_manual(breaks = sort(heatmap_color),
values = rev(scales::viridis_pal(begin = 0, end = 1)
(length(heatmap_color)))) +
theme_bw() + # make the background white
theme(panel.border = element_blank(), panel.grid.major = element_blank(),
panel.grid.minor = element_blank(), axis.ticks = element_blank(),
# erase tick marks and labels
axis.text.x = element_blank(), axis.text.y = element_blank())
return (heatmap)
}
make_cor_heatmap <- function(dat_mat, cor_mat_list){
fun_lable <- c("Pearson's Correlation", "Spearman's Correlation", "Kendall's Correlation",
"Hoeffding's D", "Blomqvist's Beta", "NMI",
"Distance Correlation", "XI Correlation")
cor_heatmap_list <- list()
cor_abs_heatmap_list <- list()
# make correlation matrices
#cor_mat_list <- make_cormat(dat_mat)
for (i in 1:8){
print(i)
if ((i == 4) | (i == 5)) {
cor_heatmap_list[[i]] <- NULL
cor_abs_heatmap_list[[i]] <- NULL
next
}
cor_mat <- abs(cor_mat_list[[i]])
# get heatmaps
cor_heatmap <- draw_heatmap(cor_mat)
# ggplot labels
ggplot_labs <- labs(title = paste("Heatmap of", fun_lable[i]),
x = "",
y = "",
fill = "Coefficient") # change the title and legend label
cor_heatmap_list[[i]] <- cor_heatmap + ggplot_labs
if (i %in% c(1,2,3,4,6)){
cor_abs_mat <- abs(cor_mat_list[[i]])
cor_abs_heatmap <- draw_heatmap(cor_abs_mat)
ggplot_abs_labs <- labs(title = paste("Abs Heatmap of", fun_lable[i]),
x = "", # change the title and legend label
y = "",
fill = "Coefficient")
cor_abs_heatmap_list[[i]] <- cor_abs_heatmap + ggplot_abs_labs
} else {
ggplot_abs_labs <- labs(title = paste("Abs Heatmap of", fun_lable[i]),
subtitle = "Equivalent to Non-Abs Heatmap",
x = "", # change the title and legend label
y = "",
fill = "Coefficient")
cor_abs_heatmap_list[[i]] <- cor_heatmap + ggplot_abs_labs
}
}
ans <- list(cor_heatmap_list, cor_abs_heatmap_list)
return (ans)
}
store_date <- date()
# save(cormat_list, heatmap_list, saver_mat, store_date,
# file ="seurat_corr_saver.RData")
# cormat_list <- make_cormat(saver_mat)
load("SAVER_beta_mat.rds")
heatmap_list <- make_cor_heatmap(sub_dat, cor_mat_list)
## [1] 1
## [1] 2
## [1] 3
## [1] 4
## [1] 5
## [1] 6
## [1] 7
## [1] 8
cor_pearson_mat <- cor_mat_list[[1]]; cor_spearman_mat <- cor_mat_list[[2]];
cor_kendall_mat <- cor_mat_list[[3]]; cor_hoeffd_mat <- cor_mat_list[[4]];
cor_blomqvist_mat <- beta_mat; cor_MI_mat <- cor_mat_list[[6]];
cor_dist_mat <- cor_mat_list[[7]]; cor_XI_mat <- cor_mat_list[[8]];
1. Pearson’s correlation coefficient
- Pearson’s correlation is to measure linear dependency of data, X and Y
- \(-1 \leq \rho_{Pearson}(X, Y) \leq 1\)
- \(\rho_{Pearson}(X, Y) = \frac{\sum(x_i-\bar{x})(y_i -\bar{y})}{\sum(x_i-\bar{x})^2(y_i -\bar{y})^2}\)
set.seed(10)
cor_pearson_mat[1:5,1:5]
## PTMA RPS5 RPS24 RPL37A RPS17
## PTMA NA NA NA NA NA
## RPS5 -0.09727472 NA NA NA NA
## RPS24 0.07422127 0.07593297 NA NA NA
## RPL37A -0.15594202 0.63299248 0.4090960 NA NA
## RPS17 -0.17080615 0.70654795 0.3664607 0.9208068 NA
quantile(cor_pearson_mat, na.rm = T)
## 0% 25% 50% 75% 100%
## -1.000000000 -0.006735407 0.020630543 0.120213354 1.000000000
quantile(abs(cor_pearson_mat), na.rm = T)
## 0% 25% 50% 75% 100%
## 9.034230e-09 1.215906e-02 4.459339e-02 1.392134e-01 1.000000e+00
# plot the smallest correlations
cor_pearson_vec <- sort(abs(cor_pearson_mat), decreasing = T)
plot(cor_pearson_vec)

#plot the high correlations
library(hexbin)
library(RColorBrewer)
high_ind <- quantile(cor_pearson_vec, probs = c(0.9), na.rm = T)
high_ind <- cor_pearson_vec[cor_pearson_vec>high_ind]
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_pearson_mat) == sample(high_ind, 1), arr.ind = T)
idx1 <- idx[1,1]; idx2 <- idx[1,2]
bin <- hexbin(sub_dat[,idx1], sub_dat[,idx2], xbins = 40)
my_colors=colorRampPalette(rev(brewer.pal(11, 'Spectral')))
plot(bin, colramp = my_colors,
xlab = paste0(colnames(sub_dat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(sub_dat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_pearson_mat[idx1, idx2], 3)))
}




#plot the lowest correlations
low_ind <- quantile(cor_pearson_vec, probs = c(0.1), na.rm = T)
low_ind <- cor_pearson_vec[cor_pearson_vec<=low_ind]
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_pearson_mat) == sample(low_ind, 1), arr.ind = T)
idx1 <- idx[1,1]; idx2 <- idx[1,2]
bin <- hexbin(sub_dat[,idx1], sub_dat[,idx2], xbins = 40)
my_colors=colorRampPalette(rev(brewer.pal(11, 'Spectral')))
plot(bin, colramp = my_colors, legend = F,
xlab = paste0(colnames(sub_dat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(sub_dat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_pearson_mat[idx1, idx2], 3)))
}




Heatmap
heatmap_list[[1]][[1]]

2. Spearman’s correlation coefficient
- It captures the monotonic relationship between data, X and Y
- \(-1 \leq \rho_{Spearman}(X,Y) \leq 1\)
- \(\rho_{Spearman} = 1 - \frac{6\sum{d_i^2}}{n(n^2-1)}\) where \(d_i\) is the difference between the ranks of \(x_i\) and \(y_i\)
cor_spearman_mat[1:5,1:5]
## PTMA RPS5 RPS24 RPL37A RPS17
## PTMA NA NA NA NA NA
## RPS5 -0.15266482 NA NA NA NA
## RPS24 0.05467073 0.03729121 NA NA NA
## RPL37A -0.19668940 0.56147422 0.3842762 NA NA
## RPS17 -0.21570545 0.64125769 0.3371530 0.9104563 NA
quantile(cor_spearman_mat, na.rm = T)
## 0% 25% 50% 75% 100%
## -1.000000000 -0.004696277 0.085982288 0.239263500 1.000000000
quantile(abs(cor_spearman_mat), na.rm = T)
## 0% 25% 50% 75% 100%
## 0.00000000 0.03577334 0.12441761 0.25469418 1.00000000
# plot the smallest correlations
cor_spearman_vec <- sort(abs(cor_spearman_mat), decreasing = T)
plot(cor_spearman_vec)

#plot the high correlations
high_ind <- quantile(cor_spearman_vec, probs = c(0.9), na.rm = T)
high_ind <- cor_spearman_vec[cor_spearman_vec>high_ind]
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_spearman_mat) == sample(high_ind, 1), arr.ind = T)
idx1 <- idx[1,1]; idx2 <- idx[1,2]
bin <- hexbin(sub_dat[,idx1], sub_dat[,idx2], xbins = 40)
my_colors=colorRampPalette(rev(brewer.pal(11, 'Spectral')))
plot(bin, colramp = my_colors, legend = F,
xlab = paste0(colnames(sub_dat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(sub_dat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_spearman_mat[idx1, idx2], 3)))
}




#plot the lowest correlations
low_ind <- quantile(cor_spearman_vec, probs = c(0.1), na.rm = T)
low_ind <- cor_spearman_vec[cor_spearman_vec<=low_ind]
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_spearman_mat) == sample(low_ind, 1), arr.ind = T)
idx1 <- idx[1,1]; idx2 <- idx[1,2]
bin <- hexbin(sub_dat[,idx1], sub_dat[,idx2], xbins = 40)
my_colors=colorRampPalette(rev(brewer.pal(11, 'Spectral')))
plot(bin, colramp = my_colors, legend = F,
xlab = paste0(colnames(sub_dat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(sub_dat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_spearman_mat[idx1, idx2], 3)))
}




Heatmap
heatmap_list[[1]][[2]]

3. Kendall’s Tau
- It is an alternative method to Spearman’s correlations, identifying monotonic relationships.
- \(-1 \leq \rho_{Kendall}(X,Y) \leq 1\)
- \(\rho_{Kendall}(X,Y) = \frac{\#\;concordant\;pairs - \#\;discordant \;pairs}{0.5n(n-1)}\)
cor_kendall_mat[1:5,1:5]
## PTMA RPS5 RPS24 RPL37A RPS17
## PTMA NA NA NA NA NA
## RPS5 -0.10267393 NA NA NA NA
## RPS24 0.03655552 0.02490967 NA NA NA
## RPL37A -0.13186234 0.39519822 0.2627274 NA NA
## RPS17 -0.14498545 0.46029968 0.2286474 0.7401223 NA
quantile(cor_kendall_mat, na.rm = T)
## 0% 25% 50% 75% 100%
## -1.000000000 -0.003906239 0.064871071 0.178346691 1.000000000
quantile(abs(cor_kendall_mat), na.rm = T)
## 0% 25% 50% 75% 100%
## 0.00000000 0.02806901 0.09346512 0.19069067 1.00000000
# plot the smallest correlations
cor_kendall_vec <- sort(abs(cor_kendall_mat), decreasing = T)
plot(cor_kendall_vec)

#plot the high correlations
high_ind <- quantile(cor_kendall_vec, probs = c(0.9), na.rm = T)
high_ind <- cor_kendall_vec[cor_kendall_vec>high_ind]
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_kendall_mat) == sample(high_ind, 1), arr.ind = T)
idx1 <- idx[1]; idx2 <- idx[2]
bin <- hexbin(sub_dat[,idx1], sub_dat[,idx2], xbins = 40)
my_colors=colorRampPalette(rev(brewer.pal(11, 'Spectral')))
plot(bin, colramp = my_colors, legend = F,
xlab = paste0(colnames(sub_dat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(sub_dat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_kendall_mat[idx1, idx2], 3)))
}




#plot the lowest correlations
low_ind <- quantile(cor_kendall_vec, probs = c(0.1), na.rm = T)
low_ind <- cor_kendall_vec[cor_kendall_vec<=low_ind]
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_kendall_mat) == sample(low_ind, 1), arr.ind = T)
idx1 <- idx[1]; idx2 <- idx[2]
bin <- hexbin(sub_dat[,idx1], sub_dat[,idx2], xbins = 40)
my_colors=colorRampPalette(rev(brewer.pal(11, 'Spectral')))
plot(bin, colramp = my_colors, legend = F,
xlab = paste0(colnames(sub_dat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(sub_dat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_kendall_mat[idx1, idx2], 3)))
}




Heatmap
heatmap_list[[1]][[3]]

4. Hoeffding’s D
- It tests the independence of data by calculating the distance between the product of the marginal distributions under the null hypothesis and the empirical bi-variate distribution.
- \(-1 \leq D(X,Y) \leq 1\)
- \(D(X,Y) = \frac{(n-2)(n-3)D_1+D_2-2(n-2)D_3}{n(n-1)(n-2)(n-3)(n-4)}\)
- \(D_1 = \sum_{i=1}^{n} Q_i(Q_i-1)\)
- \(D_2 = \sum_{i=1}^{n} (R_i-1)(R_i-2)(S_j-1)(S_j-2)\)
- \(D_3 = \sum_{i=1}^{n} (R_i-2)(S_i-2)Q_i\)
5. Blomqvist’s Beta
- It measures dependency between variables by constructing a two-way contingency table with the medians of each margin as cutting points.
- \(0 \leq \beta \leq 1\)
- \(\beta_n = \frac{n_1-n_2}{n_1+n_2} = \frac{2n_1}{n_1+n_2} - 1\)
- \(\beta = P\{(X-\tilde{x})(Y-\tilde{y})>0\} - P\{(X-\tilde{x})(Y-\tilde{y}) < 0\}\)
cor_blomqvist_mat[1:5,1:5]
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1.0000000 0.9996054 0.9957819 1.0008901 1.0009946
## [2,] 0.9996054 1.0000000 0.9973966 1.0005082 1.0004887
## [3,] 0.9957819 0.9973966 1.0000000 1.0006723 1.0007602
## [4,] 1.0008901 1.0005082 1.0006723 1.0000000 0.9979941
## [5,] 1.0009946 1.0004887 1.0007602 0.9979941 1.0000000
quantile(cor_blomqvist_mat, na.rm = T)
## 0% 25% 50% 75% 100%
## -6353.166667 -158.984277 -31.308420 -6.573932 8889.888889
quantile(abs(cor_blomqvist_mat), na.rm = T)
## 0% 25% 50% 75% 100%
## 9.665521e-06 7.453301e+00 3.322591e+01 1.617147e+02 8.889889e+03
# plot the smallest correlations
cor_blomqvist_vec <- sort(abs(cor_blomqvist_mat), decreasing = T)
plot(cor_blomqvist_vec)

#plot the high correlations
high_ind <- quantile(cor_blomqvist_vec, probs = c(0.9), na.rm = T)
high_ind <- cor_blomqvist_vec[cor_blomqvist_vec>=high_ind]
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_blomqvist_mat) == sample(high_ind, 1), arr.ind = T)
idx1 <- idx[1,1]; idx2 <- idx[1,2]
bin <- hexbin(sub_dat[,idx1], sub_dat[,idx2], xbins = 40)
my_colors=colorRampPalette(rev(brewer.pal(11, 'Spectral')))
plot(bin, colramp = my_colors, legend = F,
xlab = paste0(colnames(sub_dat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(sub_dat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_blomqvist_mat[idx1, idx2], 3)))
}




#plot the lowest correlations
low_ind <- quantile(cor_blomqvist_vec, probs = c(0.1), na.rm = T)
low_ind <- cor_blomqvist_vec[cor_blomqvist_vec<=low_ind]
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_blomqvist_mat) == sample(low_ind, 1), arr.ind = T)
idx1 <- idx[1,1]; idx2 <- idx[1,2]
bin <- hexbin(sub_dat[,idx1], sub_dat[,idx2], xbins = 40)
my_colors=colorRampPalette(rev(brewer.pal(11, 'Spectral')))
plot(bin, colramp = my_colors, legend = F,
xlab = paste0(colnames(sub_dat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(sub_dat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_blomqvist_mat[idx1, idx2], 3)))
}




Heatmap
heatmap_list[[1]][[5]]
## NULL
6. Normalized Mutual Information
- It measures how much one random variable gives information about the other. For example, High mutual information indicates a large reduction in uncertainty.
- \(0 \leq MI(X,Y) \leq 1\), as it is normalized.
- $MI(X,Y) = f_{X,Y} (x,y) log_2 ; dxdy $
- \(MI(X,Y) = \sum \sum p_{X,Y} (x,y) log \frac{p_{X,Y} (x,y)}{P_X(x)P_Y(y)}\)
cor_MI_mat[1:5,1:5]
## [,1] [,2] [,3] [,4] [,5]
## [1,] NA NA NA NA NA
## [2,] 0.07080326 NA NA NA NA
## [3,] 0.02739069 0.05052604 NA NA NA
## [4,] 0.07032199 0.19680495 0.09024883 NA NA
## [5,] 0.07149862 0.27005785 0.09091991 0.3858145 NA
quantile(cor_MI_mat, na.rm = T)
## 0% 25% 50% 75% 100%
## 0.0002222295 0.0315687819 0.0877562722 0.2224968631 1.0000000000
# plot the smallest correlations
cor_MI_vec <- sort(abs(cor_MI_mat), decreasing = T)
plot(cor_MI_vec)

#plot the high correlations
high_ind <- quantile(cor_MI_vec, probs = c(0.9), na.rm = T)
high_ind <- cor_MI_vec[cor_MI_vec>high_ind]
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_MI_mat) == sample(high_ind, 1), arr.ind = T)
idx1 <- idx[1,1]; idx2 <- idx[1,2]
bin <- hexbin(sub_dat[,idx1], sub_dat[,idx2], xbins = 40)
my_colors=colorRampPalette(rev(brewer.pal(11, 'Spectral')))
plot(bin, colramp = my_colors, legend = F,
xlab = paste0(colnames(sub_dat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(sub_dat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_MI_mat[idx1, idx2], 3)))
}




#plot the lowest correlations
low_ind <- quantile(cor_MI_vec, probs = c(0.1), na.rm = T)
low_ind <- cor_MI_vec[cor_MI_vec<=low_ind]
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_MI_mat) == sample(low_ind, 1), arr.ind = T)
idx1 <- idx[1,1]; idx2 <- idx[1,2]
bin <- hexbin(sub_dat[,idx1], sub_dat[,idx2], xbins = 40)
my_colors=colorRampPalette(rev(brewer.pal(11, 'Spectral')))
plot(bin, colramp = my_colors, legend = F,
xlab = paste0(colnames(sub_dat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(sub_dat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_MI_mat[idx1, idx2], 3)))
}




Heatmap
heatmap_list[[1]][[6]]

7. Distance Correlation
- it is measure to identify non-linear relationships between two random variables with energy distances.
- distance correlation is calculated by dividing the distance covariance between X and Y by the product of their distance standard deviations.
- \(0 \leq dCor \leq 1\)
- \(dCor(X,Y) = \frac{dCov(Y,Y)}{\sqrt{dVar(X)dVar(Y)}}\)
- \(dCov(X, Y) = \sqrt{\frac{1}{n^2} \sum_{k=1, l=1}^{n} A_{k,l}B_{k,l}}\)
- \(dVar(X) = dCov(X,X) and dVar(Y) = dCov(Y, Y)\)
cor_dist_mat[1:5,1:5]
## [,1] [,2] [,3] [,4] [,5]
## [1,] NA NA NA NA NA
## [2,] 0.021415666 NA NA NA NA
## [3,] 0.003142149 0.004832672 NA NA NA
## [4,] 0.034461122 0.291511184 0.12930026 NA NA
## [5,] 0.041251873 0.381817622 0.09936682 0.7929289 NA
quantile(cor_dist_mat, na.rm = T)
## 0% 25% 50% 75% 100%
## 0.000000000 0.001485644 0.010227512 0.041820003 0.911987832
# plot the smallest correlations
cor_dist_vec <- sort(abs(cor_dist_mat), decreasing = T)
plot(cor_dist_vec)

#plot the high correlations
high_ind <- quantile(cor_dist_vec, probs = c(0.9), na.rm = T)
high_ind <- cor_dist_vec[cor_dist_vec>high_ind]
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_dist_mat) == sample(high_ind, 1), arr.ind = T)
idx1 <- idx[1]; idx2 <- idx[2]
bin <- hexbin(sub_dat[,idx1], sub_dat[,idx2], xbins = 40)
my_colors=colorRampPalette(rev(brewer.pal(11, 'Spectral')))
plot(bin, colramp = my_colors, legend = F,
xlab = paste0(colnames(sub_dat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(sub_dat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_dist_mat[idx1, idx2], 3)))
}




#plot the lowest correlations
low_ind <- quantile(cor_dist_vec, probs = c(0.1), na.rm = T)
low_ind <- cor_dist_vec[cor_dist_vec<=low_ind]
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_dist_mat) == sample(low_ind, 1), arr.ind = T)
idx1 <- idx[1,1]; idx2 <- idx[1,2]
bin <- hexbin(sub_dat[,idx1], sub_dat[,idx2], xbins = 40)
my_colors=colorRampPalette(rev(brewer.pal(11, 'Spectral')))
plot(bin, colramp = my_colors, legend = F,
xlab = paste0(colnames(sub_dat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(sub_dat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_dist_mat[idx1, idx2], 3)))
}




Heatmap
heatmap_list[[1]][[7]]

8. Chatterjee’s XI Correlation
- It measures the degree of dependence between the variables with concept of rank.
- \(0 \leq \xi_n \leq 1\)
- \(\xi_n(X,Y) = 1 - \frac{3\sum_{i=1}^{n-1} |r_{i+1} -r_i|}{n^2-1}\)
- \(\xi_n(X,Y) = 1 - \frac{n\sum_{i=1}^{n-1}|r_{i+1}-r_i}{2\sum_{i=1}^{n}l_i(n-l_i)}\)
cor_XI_mat[1:5,1:5]
## [,1] [,2] [,3] [,4] [,5]
## [1,] NA NA NA NA NA
## [2,] 0.01928156 NA NA NA NA
## [3,] -0.01089311 -0.008368139 NA NA NA
## [4,] 0.02582162 0.200651395 0.08444043 NA NA
## [5,] 0.02491406 0.275113559 0.07579901 0.6121926 NA
quantile(cor_XI_mat, na.rm = T)
## 0% 25% 50% 75% 100%
## -0.054723119 0.002086095 0.015836627 0.048754506 0.994614676
# plot the smallest correlations
cor_XI_vec <- sort(abs(cor_XI_mat), decreasing = T)
plot(cor_XI_vec)

#plot the high correlations
high_ind <- quantile(cor_XI_vec, probs = c(0.9), na.rm = T)
high_ind <- cor_XI_vec[cor_XI_vec>high_ind]
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_XI_mat) == sample(high_ind, 1), arr.ind = T)
idx1 <- idx[1]; idx2 <- idx[2]
bin <- hexbin(sub_dat[,idx1], sub_dat[,idx2], xbins = 40)
my_colors=colorRampPalette(rev(brewer.pal(11, 'Spectral')))
plot(bin, colramp = my_colors, legend = F,
xlab = paste0(colnames(sub_dat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(sub_dat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_XI_mat[idx1, idx2], 3)))
}




#plot the lowest correlations
low_ind <- quantile(cor_XI_vec, probs = c(0.1), na.rm = T)
low_ind <- cor_XI_vec[cor_XI_vec<=low_ind]
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_XI_mat) == sample(low_ind, 1), arr.ind = T)
idx1 <- idx[1]; idx2 <- idx[2]
bin <- hexbin(sub_dat[,idx1], sub_dat[,idx2], xbins = 40)
my_colors=colorRampPalette(rev(brewer.pal(11, 'Spectral')))
plot(bin, colramp = my_colors, legend = F,
xlab = paste0(colnames(sub_dat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(sub_dat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_XI_mat[idx1, idx2], 3)))
}




Heatmap
heatmap_list[[1]][[8]]

quantile_mat <- c()
for (i in 1:length(cor_mat_list)){
if (i == 4) {
quantile_mat <- rbind(quantile_mat, c(NA, NA))
next
}
quantile_mat <- rbind(quantile_mat,
quantile(abs(cor_mat_list[[i]]), probs = c(0.15, 0.85), na.rm = T))
}
rownames(quantile_mat) <- c("Pearson", "Spearman", "Kendall", "Hoeffding's D",
"Blomqvist's Beta","Dist. Corr", "NMI", "XI Corr")
quantile_mat
## 15% 85%
## Pearson 0.0063573876 0.20926968
## Spearman 0.0175490367 0.33646081
## Kendall 0.0143049742 0.25347375
## Hoeffding's D NA NA
## Blomqvist's Beta 0.9893333333 1.00000000
## Dist. Corr 0.0173682245 0.35759836
## NMI 0.0005631287 0.07361977
## XI Corr 0.0031408643 0.08356398
# save(quantile_mat, store_date, file = "CD4_seurat_corrq.RData")